How many lockers are closed? problem

A high school has 1,000 students and it has 1,000 lockers in its main corridor. All of the lockers are initially open. The first student walks down the corridor and changes the position (open/close) of each locker. In his case, he closes all of the lockers since they are all open. Then the second student walks down the corridor and changes the position of every second locker (in his case he will open all of the even-numbered lockers). Then the third student walks down the corridor and changes the position of every third locker (in his case he opens some and closes some). Then the fourth student walks down and changes the position of every fourth locker, etc. How many lockers are closed after the one-thousandth student is done? (Remember that the last student changes the position of every one-thousandth locker, which means he only changes the position of the last locker.)

Re: How many lockers are closed? problem

I don't know. I just thought I get this started.

1 is closed.
I think all the prime number ones will be open. (2 factors)
Any with 3 factors less than 1000 will be closed
Any with 4 factors less than 1000 will be open
So if it has an even number of factors less than 1000 it will be open
If it has an odd number of factors less than 1000 it will be closed

I am making this up as I go. It is probably all wrong and it probably doesn't help anyway.

Re: How many lockers are closed? problem

I thought it was all the perfect squares (so like 31?).. but that's if you start with them closed. Here we start with them open

Originally Posted by Melody2

I don't know. I just thought I get this started.

1 is closed.
I think all the prime number ones will be open. (2 factors)
Any with 3 factors less than 1000 will be closed
Any with 4 factors less than 1000 will be open
So if it has an even number of factors less than 1000 it will be open
If it has an odd number of factors less than 1000 it will be closed

I am making this up as I go. It is probably all wrong and it probably doesn't help anyway.

Re: How many lockers are closed? problem

Originally Posted by fifthrapiers

A high school has 1,000 students and it has 1,000 lockers in its main corridor. All of the lockers are initially open. The first student walks down the corridor and changes the position (open/close) of each locker. In his case, he closes all of the lockers since they are all open. Then the second student walks down the corridor and changes the position of every second locker (in his case he will open all of the even-numbered lockers). Then the third student walks down the corridor and changes the position of every third locker (in his case he opens some and closes some). Then the fourth student walks down and changes the position of every fourth locker, etc. How many lockers are closed after the one-thousandth student is done? (Remember that the last student changes the position of every one-thousandth locker, which means he only changes the position of the last locker.)

Re: How many lockers are closed? problem

Wow. I came up with something that was relevant. How about that!

Ofcourse,

the light just came on.
Factors come in pairs. The only time when there will be an odd number of factors is when 2 of them are the same.
That is, it would have to be a square number.
So the square number lockers will be closed
31sqared = 961
so 31 will be closed. The rest will be opend.

I know, you are giggling at me, the page Plato sent me to probably said all this. But I am pleased with myself anyway.